Child care is definitely one of (many different) reasons that people use private schools.
If the state school doesn’t have wrap around care then options may be very limited. Even if it does have wrap around care it may be of poor quality. Behaviour standards were much lower than during the school day and kids who were physically bullied asked to leave at ours. No suggestion of asking the bullies to leave though.
There are no childminders in some areas (high house prices), au pairs are no longer an option due to Brexit so you are left with a nanny. Only full time is now an option as understandably people need to earn a living and childcare is in short supply but the cost of that is easily more than two sets of prep fees plus wrap around.
I would (and have) choosen to be a SAHM in simar circumstances even though I could earn a salary that is similar to a senior doctor. As almost all of the professional women, lawyers, doctors, accountants, academics etc around me have made a similar choice at some point for a period of time at least.
I totally appreciate this is a small proportion of the world that this impacts though. And I am not suggesting this needs sympathy for people who can afford to stay at home. But to suggest that professional salaries mean everyone can or find or would be willing to take up childcare solutions available without private schools is wrong.
I agree it is unlikely that many parents will move kids and drop out of working to become a SAHM. But higher fees may shift the calculations for future parents in terms of returning to work or continuing work post the nursery years.